Research Log: Week 2 |
| Monday, 5/23/05 |
| Today, we found the tangent line to the flat version of the Knight's visor. At first we tried to take the derivative of the implicit form of the curve and obtained unnecessarily complicated equations that also gave the wrong graph. Then Joe pointed out that we could just take the derivative of the parametric form. This was very simple and gave the right answer. Then, we noticed that, since the tangent line to the curve is a linear approximation of the curve, and the curve is determined by reflecting each point on the x-axis, call it a, across the tangent line to the circle at that point, we could estimate the tangent line to the curve by reflecting $a + \Delta a$ across the tangent line at a. This would mean that the tangent line to the curve is the reflection of the x-axis across the tangent line to the circle at a. It would also mean, as we had hypothesized, that the tangent line was perpendicular to the rib to which it was attached. The intuition, however, was not a proof. Together with Joe, we proved that the tangent line to the curve was indeed perpendicular to its rib by dotting the equations of the two direction vectors together. |
| Tuesday, 5/24/05 |
| We made a Mathematica file that graphed all of our computed curves for the flat version of the Knight's visor together. Then we went to the computer lab and learned Adobe Illustrator, and also graphed a picture of the intersection of the two planes and sphere mentioned in last week's log. |
| Wednesday, 5/25/05 |
| Today, Duc and I added to Joe's Latex file. Then we edited our picture in Adobe Illustrator. |
| Thursday, 5/26/05 |
| After graphing and editing pictures for the 3D version of the Knight's visor and after Joe had spent some time using Adobe Illustrator to fix the pictures to include in his book, we realized that we had made a mistake in graphing one of the planes in our two plane, one sphere intersection. We spent most of the day fixing the picture in Illustrator, and a few minutes editing the accompanying text. |
| Friday, 5/27/05 |
| I went to the ophthalmologist in the morning while Duc finished editing in Illustrator. When I got back, we wrote the equation of the tennis ball-like racetrack D-form, and graphed it in Mathematica. We didn't use curvature and torsion since the equation of the curve itself was even more straightforward to calculate than the equations of curvature and torsion. |